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Proof step 27 from proof inv-close (previous step)

of[fd] = mkfile(ROOT_INO, mode, m0), files[ROOT_INO] = mkfile(m1, md, m2, content), dom of < ino0, afs-cons(dirs, files), of-cons(of -- fd, dirs, files), of-cons(of, dirs, files), fd ∈ of ⊦ is-open(ROOT_INO, of -- fd)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 28:
⊦ of-cons(of0, dirs0, files0) → (of-cons(of1, dirs1, files1) ↔ (∀ n. (n ∈ of0 → of0[n].ino ∈ files0) → n ∈ of1 → of1[n].ino ∈ files1))
Proof step 29:
files[ROOT_INO] = mkfile(m1, md, m2, content), of[fd] = mkfile(ROOT_INO, mode, m0), dom of < ino0, afs-cons(dirs, files), of-cons(of, dirs, files), fd ∈ of, ∀ n. (n ∈ of → of[n].ino ∈ files) → n ∈ of -- fd → (of -- fd)[n].ino ∈ files ⊦ is-open(ROOT_INO, of -- fd)

Rule argument

spec: <toplevel> name: of-cons-reduce seq: ⊦ of-cons(of0, dirs0, files0) → ( of-cons(of1, dirs1, files1) ↔ (∀ n. (n ∈ of0 → of0[n].ino ∈ files0) → n ∈ of1 → of1[n].ino ∈ files1)) subst: [of0, dirs0, files0, files1, dirs1, of1] → [of, dirs, files, files, dirs, of -- fd] flags: <all paths>

Additional information

Used terms: of, dirs, files, files, dirs, of -- fd